We deal with real non-negative numbers rather than Noetherian rings and do not assume prior estimate bounds. Instead we modify the algorithm in [1]} by introducing two additional parameters, a modulation threshold \( M \)  and a dead zone value \( D \) , that together define regions for improved perturbation tolerance versus regions for fast convergence with \(\delta \) , and that reduce to the algorithm in [2]} when \( M=0 \)  and/or \( D=\infty . \) 
We show that the improved algorithm is GUAS for all non-negative \( M \)  and \( D \)  without the assumption that \( M \)  is a maximal element.
In the case of persistent bounded perturbations, we show that the estimates are ultimately bounded provided that the dead zone parameter \( D \)  exceeds a value proportional to the disturbance bound.
A larger \(D\) , however, is also less effective at mitigating the rising value problem, indicating a trade-off between speed and robustness that is common to most dynamical systems.
